2 Problem: why is the sawtooth crash so fast at large Lundquist number S=t R /t A? t R = a 2 /h t A = a/v Alfven Resistive time Alfven time

3 The m=1 mode and the sawtooth crash The m=1 mode: an essential element of the sawtooth cycle [Von Goeler et al.(1974), Kadomtsev(1975)] Process mediated by the restivity h. Becomes slow at large S, small h. The sawtooth crash time in JET is shorter than the collision time. [Edwards et al., (1986)] This process is too fast to be explained by resistive MHD Non-collisional physics (electron inertia and other effects) gives the right time scale (100 microseconds) for JET: γ τ A ~ (d e /L) both in the linear and in the nonlinear phase. (d e = skin depth, L ~a~r = machine scale length. τ A = Alfven time). Typically d e /L~10-3 [Ottaviani and Porcelli, (1993)]. Probably not sufficient to explain fast crashes in medium size tokamaks and in some simulations at moderate Lundquist number

4 Look for a new explanation

5 What about the stability of m=1 islands? Current sheets of large aspect ratio are developed during the nonlinear stage of primary internal-kink modes. How does their instability to secondary reconnecting modes relate to fast sawtooth crashes? Ex : Δ' = ψ J φ ( Figures taken from [Q.Yu et al., Nucl. Fusion 54, (2014)] )

6 What about the stability of m=1 islands? Current sheets of large aspect ratio are developed during the nonlinear stage of primary internal-kink modes. How does their instability to secondary reconnecting modes relate to fast sawtooth crashes? Ex : Δ' = ψ J φ ( Figures taken from [Q.Yu et al., Nucl. Fusion 54, (2014)] ) CURRENT SHEET

7 What about the stability of m=1 islands? Current sheets of large aspect ratio are developed during the nonlinear stage of primary internal-kink modes. How does their instability to secondary reconnecting modes relate to fast sawtooth crashes? Ex : Δ' = ψ J φ Secondary islands ( Figures taken from [Q.Yu et al., Nucl. Fusion 54, (2014)] )

10 Slab approximation for the current sheet B 0 L a ~ δ I Due to the current sheet's large aspect ratio L/δ I >> 1 a quasi-continuum spectrum of unstable wavenumbers can be assumed. The maximum growth rate occurs at short wavelength. This justifies the local assumption about the geometry

11 Key assumptions 1) In the current sheet, the current density is comparable to the equilibrium current density which is what one gets at the end of the linear phase of the m=1 mode: J cs ~ J 0 2) The above is reasonable if the instability of the current sheet (what we call the secondary instability) is fast enough that its growth rate exceeds the evolution rate of the equilibrium (i.e. the primary reconnecting mode): γ II > τ cs -1 ~ γ I This is verified a posteriori. The m=1 island can be considered as static. 3) The width of the current sheet is assumed to be of the order of the m=1 layer width. This is justified at the end of the linear phase/beginning of the nonlinear phase of the m=1 primary island 4) We perform an asymptotic analysis (i.e. Lundquist number S, etc. )

12 Size of the transverse magnetic field in the current sheet In a tokamak L 0 ~ R and L ~ R ~ L 0. And the end of the internal-kink linear phase a ~ δ I = S -1/3 L B 0 Example of Harris-pinch equilibrium, typical for a / L ka << 1 (thus valid for the tearing at small ka and for the dominant mode at Δ'δ ~ 1) L ~ L 0 Note: in several papers B CS ~ B 0 and J CS >> J 0 NOT JUSTIFIED a width ~ L ~ L 0

26 Numerical simulations (ongoing work) current current current current t=301 t=321 t=325 t=331

27 Numerical simulations (ongoing work) current current current current t=301 t=321 t=325 t=331 Current sheet instability and splitting

28 Numerical simulations (ongoing work) current current current current t=301 t=321 t=325 t=331 Current maxima Current sheet instability and splitting absolute maximum value at primary X-point time

29 Numerical simulations (ongoing work) current current current current t=301 t=321 t=325 t=331 Current maxima Current sheet instability and splitting absolute maximum value at primary X-point Pseudo-spectral simulations Box aspect ratio A=2 Lundquist number S=10 3 Still work to do time

30 Summary and conclusions A current sheet generated by a primary instability with Δ'=, such as the resistive internal kink mode, becomes unstable at an early stage in its nonlinear development because of sub- Alfvenic modes. In the case of sawtooth phenomenon in a purely resistive framework the current sheet becomes unstable to a secondary mode with growth rate γτ A ~ S -1/6, apparently in agreement with the numerical results of [Yu et al., Nucl. Fusion 54, (2014)]. When finite electron inertia is included in the analysis, for (L/d e ) 12/5 < S < (L/d e ) 3, regime relevant to most medium size tokamak devices, the secondary instability develops in the inertiadriven collisionless regime with a growth rate γ II τ A ~ (d e /a) 2 ~ (d e /L) 2 S 2/3 which can become near-alfvenic. The overall reconnection rate is at least of order γ τ A ~(d e /L)2/5, always larger than the internal kink growth rate. Paper: D. Del Sarto, M. Ottaviani Secondary fast instability in the sawtooth crash, Physics of Plasmas 24, (2017)

31 THANK YOU FOR YOUR ATTENTION

32 On the plasmoid scaling A «plasmoid» scaling of superfast (faster than Alfvenic) reconnection rate of current sheets has been proposed in the literature (Loureiro et al., 2005, and subsequent studies) γ plasmoid τ A ~ S 1/4 This scaling requires two conditions (Tajima and Shibata, Space Astrophysics, 2002) 1. An intense current sheet such that B CS ~ B 0 and J CS / J 0 ~ L/d sheet >> 1 AND 2. A sheet size such that d sheet /L ~ S -1/2 As we have seen, this is not justified in the sawtooth case Current sheets such as d sheet /L ~ S -1/3 and J CS / J 0 ~ L/d sheet produce Alfvenic rate reconnection such that γ τ A ~ 1 which is a situation of likely interest in astrophysics where sheets can be produced by Alfvenic motion (Pucci and Velli, 2014)

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